[Paper Review] Stable vector bundles on a hyper-Kahler manifold with a rank 1 obstruction map are modular

Eyal Markman|arXiv (Cornell University)|Jul 29, 2021

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper establishes that slope-stable reflexive sheaves on hyper-Kähler manifolds with a rank 1 obstruction map to deformations of the derived category are 'very modular,' meaning their endomorphism algebras deform with the manifold. The key result shows that such sheaves define a line in the Looijenga-Lunts-Verbitsky lattice, and three main sources of examples are constructed: via derived equivalences of Hilbert schemes, images of sky-scraper sheaves, and line bundles on Lagrangian subvarieties with rational canonical bundles.

ABSTRACT

Let X be an irreducible 2n-dimensional holomorphic symplectic manifold. A reflexive sheaf F is very modular, if its Azumaya algebra End(F) deforms with X to every Kahler deformation of X. We show that if F is a slope-stable reflexive sheaf of positive rank and the obstruction map from the second Hochschild cohomology of X to $Ext^2(F,F)$ has rank 1, then F is very modular. We associate to such a sheaf a vector in the Looijenga-Lunts-Verbitsky lattice of rank equal to the second Betti number of X plus 2. Three sources of examples of such modular sheaves emerge. The first source consists of slope-stable reflexive sheaves F of positive rank which are isomorphic to the image of the structure sheaf via an equivalence of the derived categories of two irreducible holomorphic symplectic manifolds. The second source consists of such F, which are isomorphic to the image of a sky-scraper sheaf via a derived equivalence. The third source consists of images of torsion sheaves L supported as line bundles on holomorphic lagrangian submanifolds Z, such that Z deforms with X in co-dimension one in moduli and L is a rational power of the canonical line bundle of Z.

Motivation & Objective

• To prove that reflexive sheaves on hyper-Kähler manifolds with a rank 1 obstruction map are very modular, i.e., their endomorphism algebras deform with the manifold in all Kähler deformations.
• To associate such sheaves with a well-defined line in the Looijenga-Lunts-Verbitsky (LLV) lattice of rank $ b_2(X) + 2 $.
• To construct three distinct families of examples of such modular sheaves using derived categories, Hilbert schemes, and Lagrangian subvarieties.
• To generalize and extend O'Grady's results on $ K3^{[2]} $-type manifolds to higher-dimensional hyper-Kähler varieties.

Proposed method

• Use the obstruction map $ \mathrm{obs}_F: HH^2(X) \to \mathrm{Ext}^2(F,F) $ to characterize sheaves with minimal deformation obstructions.
• Define 'very modular' sheaves as those for which $ \mathcal{E}nd(F) $ deforms with every Kähler deformation of $ X $.
• Construct a canonical line in the LLV lattice via the image of the obstruction map, using the Hochschild cohomology and derived category deformations.
• Apply the BKR correspondence to lift equivariant bundles on $ X^n $ to sheaves on the Hilbert scheme $ X^{[n]} $, ensuring reflexivity and modularity.
• Use derived equivalences $ \Phi: D^b(X) \to D^b(Y) $ to transfer known modular objects (e.g., structure sheaf, sky-scraper sheaves) to new examples on $ X^{[n]} $.
• Analyze the Chern character and Mukai vector of Fourier-Mukai transforms to verify integrality and positivity conditions for modularity.

Experimental results

Research questions

• RQ1 Under what conditions does a reflexive sheaf $ F $ on a hyper-Kähler manifold $ X $ deform with every Kähler deformation of $ X $?
• RQ2 When does the obstruction map $ \mathrm{obs}_F: HH^2(X) \to \mathrm{Ext}^2(F,F) $ having rank 1 imply that $ F $ is very modular?
• RQ3 What are the geometric and categorical sources of stable reflexive sheaves with rank 1 obstruction maps on $ X^{[n]} $-type hyper-Kähler manifolds?
• RQ4 How can the Looijenga-Lunts-Verbitsky lattice be used to classify and construct such modular sheaves?
• RQ5 Can the modularity of a sheaf be detected via its Chern character and the action of the derived monodromy group?

Key findings

• A slope-stable reflexive sheaf $ F $ on a hyper-Kähler manifold $ X $ with $ \mathrm{rank}(\mathrm{obs}_F) = 1 $ is very modular, i.e., $ \mathcal{E}nd(F) $ deforms with $ X $ to every Kähler deformation.
• Such a sheaf determines a well-defined line in the LLV lattice of $ X $, which is preserved under derived autoequivalences.
• The first source of examples arises from $ \Phi(\mathcal{O}_X) $, where $ \Phi: D^b(X) \to D^b(Y) $ is a derived equivalence between two irreducible holomorphic symplectic manifolds.
• The second source consists of $ \Phi(\mathcal{O}_x) $, the image of a sky-scraper sheaf under a derived equivalence, yielding very modular sheaves on $ X^{[n]} $.
• The third source includes $ \Phi(L) $, where $ L $ is a rational power of the canonical bundle on a Lagrangian subvariety $ Z \subset X $ that deforms in codimension one in moduli.
• An explicit construction is given for $ X = S^{[n]} $, where $ S $ is a $ K3 $ surface, using the BKR correspondence applied to $ G^{oxtimes n} $, yielding a very modular bundle on $ S^{[n]} $.